Allardyce saw his side surrender a 2-1 lead on Saturday after Mark Noble was dismissed with 10 minutes remaining, extending their winless run to four matches.
He was less than impressed with referee Lee Mason and urged the Football Association to relax their stance towards managers who question match officials and their decisions in their post-match interviews.
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"I've got to be careful what I say, unfortunately," he said.
"We all have to be careful because we get brought up about what we say about the referee and we get fined.
"What I can say about it is; do I agree with that? No. Why? Because that's the wrong sort of protection for referees as far as I'm concerned.
"If we want to make the referees better then they've got to be criticised like us all.
"We all get criticised in this game but for me to say too much about how I felt the referee was today then I would get myself into trouble and the last thing I want to do is give the FA any of my money that I work really hard for.
"I will take it up with a delegate.
"I'll relook at the performance of my team and the referee tomorrow [Sunday] and make the report to the delegate how I see fit.
"That's all I can do within the system we have today."
West Ham led twice against Everton, with Ravel Morrison and Noble scoring, but a double from Leighton Baines and Romelu Lukaku's debut strike handed the visitors the points.